Finite Group Databasebeta

← #29690570

Subgroups — #29690570

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839817C211yesmaximal, characteristic
2#21448224C211yesmaximal, characteristic
3#382×D61C211yesmaximal, characteristic
4#21446590C212yesmaximal, complemented
5#21446586C212yesmaximal, complemented
6#21446013C212yesmaximal
7#21446008C212yesmaximal
8#21445988C211yesmaximal, characteristic, complemented
9#21445987C211yesmaximal, characteristic, complemented
10#21445954C211yesmaximal, characteristic, complemented
11C61×#1836C211yesmaximal, characteristic, complemented
12C2×#14519735—21no—
13C2×C4⋊C4×D61C2211yescharacteristic
14C2×C4⋊C4×D61C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16839817C418yescomplemented
17C2×#14519711C2212yes—
18C2×#14519710C2212yes—
19C2×#14519709C2212yes—
20C2×#14519708C2212yes—
21C2×#14519706C2212yes—
22C2×#14519689C2212yes—
23C2×#14519688C2211yescharacteristic
24#16839632C2211yescharacteristic
25#16839628C2211yescharacteristic
26#16839627—21no—
27#16839627C2211yescharacteristic
28#16839627C2211yescharacteristic
29#16839626—21no—
30#16839626—21no—
31#16839619—21no—
32#16839619C2212yes—
33C4⋊C4×Dic61—21no—
34C4⋊C4×Dic61C2212yes—
35#95×D61—21no—
36#95×D61—21no—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×#97×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#382C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C22×C4×D61—21no—
45C22×C4×D61—21no—
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49C22×#92418C2312yes—
50#14519735—44no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2312yes—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C4⋊C4×D61—42no—
59C4⋊C4×D61—42no—
60C2×#11758198—22no—
61C2×C4×Dic61—22no—
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61D411yescharacteristic
64C2×C4×Dic61C2311yescharacteristic
65C42×D61C2×C414yes—
66C42×D61Q814yes—
67#14519711—42no—
68#14519710—42no—
69#14519709—42no—
70#14519708—42no—
71#14519706C2×C418yescomplemented
72#14519689C2×C418yes—
73#14519688C2×C414yescomplemented
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79#14519684—21no—
80#14519684—22no—
81#14519684—22no—
82C22×C8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89#97×C61C2×C414yescomplemented
90#95×C61—21no—
91#95×C61—21no—
92C23×D61C2×D411yescharacteristic
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2411yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61C2×D411yescharacteristic
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×C4×D61C4⋊C414yes—
118C2×C4×D61C4⋊C414yes—
119C2×C4×D61C22×C412yes—
120C2×C4×D61C2×Q812yes—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61C22×C414yes—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61—42no—
128C4⋊C4×C61—42no—
129C42×C61C22×C412yes—
130C42×C61C2×Q812yes—
131#11758198—44no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136C4⋊Dic61—42no—
137C4⋊Dic61—42no—
138#11758196—42no—
139#11758196—42no—
140C4×Dic61—44no—
141C4×Dic61C4⋊C414yes—
142C4×Dic61C4⋊C414yes—
143C4×Dic61C22×C412yes—
144C4×Dic61C2×Q812yes—
145C2×#92418—22no—
146C2×#92418—22no—
147C2×#92418C22×C414yes—
148C23×C61C22×D411yescharacteristic
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—21no—
153C22×D61C2×C4⋊C412yes—
154C22×D61AGL(1,ℤ/8)12yes—
155C22×D61#13712yes—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61C2×C4⋊C412yes—
170C2×C4×C61C2×C4⋊C412yes—
171C2×C4×C61C23×C411yescharacteristic
172C2×C4×C61C22×Q811yescharacteristic
173C2×C4×C61H(5,𝔽2)12yes—
174C2×C4×C612−1+412yes—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—21no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61C2×C4⋊C412yes—
191C2×Dic61C2×C4⋊C412yes—
192C2×Dic61C2×C4⋊C412yes—
193C4×D61—28no—
194C4×D61—24no—
195C4×D61—24no—
196C4×D61—44no—
197C4×D61—84no—
198C4×D61—84no—
199C8×C61—24no—
200#92418—24no—
201#21986—611nomaximal, Sylow
202C22×C61C22×C4⋊C411yescharacteristic
203C22×C61#53511yescharacteristic
204C22×C61#53611yescharacteristic
205C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
206C22×C61C2×#13711yescharacteristic
207C22×C61#57511yescharacteristic
208C22×C61#57511yescharacteristic
209C2×D61—22no—
210C2×D61—22no—
211C2×D61—22no—
212C2×D61—22no—
213C2×D61—22no—
214C2×D61—22no—
215C2×D61#42814yes—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61—82no—
221C4×C61#52012yes—
222C4×C61#54412yes—
223Dic61—22no—
224Dic61—22no—
225Dic61—22no—
226Dic61—81no—
227Dic61—82no—
228Dic61—81no—
229C2×#432—611no—
230#2285—611no—
231C2×#382—611no—
232#1949—612no—
233#1949—612no—
234#1853—612no—
235#1852—612no—
236#1847—611no—
237#1847—611no—
238#1836—611no—
239#1836D6111yescharacteristic, complemented
240C2×C61#231711yescharacteristic
241C2×C61C2×#42811yescharacteristic
242C2×C61#291111yescharacteristic
243C2×C61#321311yescharacteristic
244C2×C61#321511yescharacteristic
245C2×C61#323411yescharacteristic
246C2×C61#323611yescharacteristic
247D61—24no—
248C2×#117—1221no—
249C22×C4⋊C4—611no—
250C22×C4⋊C4—611no—
251C22×C42—611no—
252#432—618no—
253C2×#107—612no—
254C2×#107—612no—
255C2×#106—612no—
256C2×#106—612no—
257C2×C4⋊C8—612no—
258C2×C22⋊C8—612no—
259C2×#97—611no—
260C2×#97C2×D6111yescharacteristic
261#384—611no—
262#384C2×D6111yescharacteristic
263#382—611no—
264#382—611no—
265#382—611no—
266#382—1221no—
267#382—1221no—
268#382—1221no—
269#382C2×D6111yescharacteristic
270#380—612no—
271#380—1221no—
272C4×C4⋊C4—612no—
273C4×C4⋊C4—1221no—
274C2×#95—1221no—
275C2×#95—1221no—
276#340—612no—
277#340—612no—
278#340C2×D6112yes—
279#340C2×D6112yes—
280C61#2198611yescharacteristic, complemented, Sylow
281C23×C4—611no—
282C23×C4—611no—
283C23×C4—611no—
284C23×C4—1221no—
285C23×C4—1221no—
286C22×C8—612no—
287C22×C8C22×D6112yes—
288#117—2444no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—611no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—612no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4—2442no—
299C2×C4⋊C4—2442no—
300C2×C4⋊C4C22×D6111yescharacteristic
301C2×C4⋊C4C22×D6111yescharacteristic
302C2×C4⋊C4C22×D6112yes—
303C2×C22⋊C4—1222no—
304C2×C42—611no—
305C2×C42—611no—
306C2×C42—611no—
307C2×C42—614no—
308C2×C42—614no—
309C2×C42—1222no—
310C2×C42C22×D6111yescharacteristic
311#107—2442no—
312#107—2442no—
313#106—2442no—
314#106—2442no—
315C4⋊C8—618no—
316C22⋊C8—618no—
317#97—614no—
318#97C4×D6114yescomplemented
319#95—21no—
320#95—21no—
321#95—1221no—
322#95—1221no—
323#95—1221no—
324#95—1221no—
325#95—1222no—
326#95—1222no—
327#95—1221no—
328#95—1221no—
329C24—611no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—611no—
334C22×C4—611no—
335C22×C4—611no—
336C22×C4—612no—
337C22×C4—612no—
338C22×C4—614no—
339C22×C4—614no—
340C22×C4—1222no—
341C22×C4—1222no—
342C22×C4—1222no—
343C22×C4—1222no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—1222no—
347C22×C4—2441no—
348C22×C4—2441no—
349C22×C4—2442no—
350C22×C4—2442no—
351C22×C4—2442no—
352C22×C4—2442no—
353C22×C4—2442no—
354C22×C4—2442no—
355C22×C4D4×D6111yescharacteristic
356C22×C4D4×D6111yescharacteristic
357C22×C4C23×D6111yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—614no—
361C2×C8—1222no—
362C2×C8—1222no—
363C2×C8C2×C4×D6114yes—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4—1222no—
372C4⋊C4—1222no—
373C4⋊C4—2442no—
374C4⋊C4—2442no—
375C4⋊C4—2442no—
376C4⋊C4—2442no—
377C22⋊C4—2444no—
378C42—612no—
379C42—612no—
380C42—614no—
381C42—614no—
382C42—2444no—
383C42C2×C4×D6112yes—
384C42Q8×D6112yes—
385C23—612no—
386C23—612no—
387C23—612no—
388C23—1221no—
389C23—1221no—
390C23—1221no—
391C23—1221no—
392C23C2×D4×D6111yescharacteristic
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—612no—
407C2×C4—612no—
408C2×C4—612no—
409C2×C4—1228no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1224no—
417C2×C4—1224no—
418C2×C4—4884no—
419C2×C4—4884no—
420C2×C4—2444no—
421C2×C4—2442no—
422C2×C4—2441no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2442no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2442no—
430C2×C4C4⋊C4×D6112yes—
431C2×C4C4⋊C4×D6112yes—
432C2×C4C22×C4×D6111yescharacteristic
433C2×C4C2×Q8×D6111yescharacteristic
434C2×C4#1451982312yes—
435C2×C4#1451982412yes—
436C8—24no—
437C8—1224no—
438C22—614no—
439C22—1222no—
440C22—1222no—
441C22—1222no—
442C22—1222no—
443C22—1222no—
444C22—1222no—
445C22C2×C4⋊C4×D6111yescharacteristic
446C22#1684033011yescharacteristic
447C22#1684033211yescharacteristic
448C22AGL(1,ℤ/8)×D6111yescharacteristic
449C22#137×D6111yescharacteristic
450C22#1684055811yescharacteristic
451C22#1684056111yescharacteristic
452C4—22no—
453C4—22no—
454C4—82no—
455C4—81no—
456C4—81no—
457C4—1222no—
458C4—1222no—
459C4—1222no—
460C4—4881no—
461C4—4882no—
462C4—4881no—
463C4#1684024212yes—
464C4#1684039212yes—
465C2—1224no—
466C2#2144844411yescharacteristic
467C2#428×D6111yescharacteristic
468C2#2145440611yescharacteristic
469C2#2145680711yescharacteristic
470C2#2145682311yescharacteristic
471C2#2145700411yescharacteristic
472C2#2145702011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1836C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1836D61
hypocenterC61#21986
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #21446590C2semidirect
5. #21446586C2semidirect
8. #21445988C2semidirect
9. #21445987C2semidirect
10. #21445954C2semidirect
11. C61×#1836C2semidirect
16. #16839817C4semidirect
71. #14519706C2×C4semidirect
73. #14519688C2×C4semidirect
89. #97×C61C2×C4semidirect
239. #1836D61semidirect
280. C61#21986semidirect
318. #97C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168398172
#214482242
#382×D612
#214465902
#214465862
#214460132
#214460082
#214459882
#214459872
#214459542
C61×#18362
#2198661